HOW BIG IS A MAXITRONIUM?
By: Tito Serrano
First written as “How Big is a Pushul?” in November 1962.
Changed to “How Big is a Maxitronium?” in 1992.
How big is a maxitronium? Let me tell you that a maxitronium is quite a lot. When you count maxitroniums even googols become insignificant. In 1940 Ed Kasner and Jim Newman wrote the book “Mathematics and the Imagination.” In “Mathematics and the Imagination” Ed and Jim defined “googol” as 10100 or a one with 100 zeros. Isn’t the name “googol” totally awful sounding? The son of either Ed or Jim proposed the name; therefore, we will restrain from leveling any criticism.
Ed and Jim introduced the “googoplex” too. A “googoplex” equals 10googol. I bet that you can write numbers that are bigger than the numbers just discussed. So can I: googoplex!googoplex!. Numbers are uninteresting unless they have meaning, are they not? These numbers are boring because they mean nothing. To make a number exiting turn it into a speed, or measure a distance with it or better yet, in front of it place a dollar sign. The usefulness of the googol eludes me, but I know exactly how much a maxitronium is. To define a maxitronium, however, I must take the long way around.
Here we go: Take a deck of 52 cards as an example. Ask yourself in how many different ways can it be shuffled? The correct answer is factorial 52 (52!). That’s about 1.6 • 1066. A factorial, if you recall your high school math, is the product of all the positive integers from one to a given number. For example, factorial 3, usually written 3! is the product of 1 • 2 • 3, or 3! = 6. Factorial 5 (5!) is the product of 1 • 2 • 3 • 4 • 5, or 5! = 120. 11! = 39,916,800
A maxitronium responds to a similar manipulation. If you pack the universe full of protons, with no empty space between them, and then you set out to figure the total number of different proton arrangements possible, the resulting number is exactly and precisely one maxitronium. This cannot be done without first knowing the volumes of the entire universe as well as the volume of a tiny proton.
To visualize the enormity of a maxitronium, we should have an idea of how big the universe is. The first thing on our agenda should be accepting, at least for now, Hubble’s constant (1929):
R = kD or D=R/k or k= R/D
where R (for rapidly) is the speed at which quasars move away from us, k is a constant, and D is the instantaneous distance between some quasar and us. To be sure, there are plenty of astronomers who don’t like Hubble’s constant, but that is the best we seem to have … for now.
Hubble believed in an expanding universe. Modern telescopic observation seems to support Hubble’s model of creation. As quasars move away from us, they display a linear increase in speed correspondent to their increase in distance from us. That means that they are speeding and the further they go the faster they speed. There is a rate for that increase in speed (acceleration) that is constant. Constant “k” gives us this acceleration constant. Some astronomers place the value of k at about 50 to 100 kilometers per second per million parsecs. In my calculations, I use the mid-point of seventy�five kilometers per second per million parsecs [(75k/s)/(1 • 106 parsecs)]. A parsec is about 3.26 light years or 30,900,000,000,000 kilometers, or 19,200,000,000,000 miles.
We know that with every increase of one million parsecs, the speed of a quasar moving away from us increases by 75 kilometers per second. We also must know from sheer logic that if we keep increasing another 75 kilometers per second again and again, the speed of light will be reached at some point. We will never see if it is possible for quasars to move faster than the speed of light, since their light would not reach us if they did. (Although secretly I know in my bones that quasars move faster than the speed of light. I further suspect that in the past light must have traveled faster than it does today, although today we call this speed (2.997925 • 108 meters per second) a constant: the electromagnetic constant!) As far as we are concerned, we have a spherical universe. The radius of this universe-sphere is the distance at which quasars surpass the speed of light: 300,000,000 meters per second (186,000 miles per second). Our visible universe if FINITE.
Going back to the equation for Hubble’s constant:
R is equal to the speed of light. We can substitute 75 kilometers per second, per million parsecs for constant k. That is all we need to find the edge of our universe-sphere. The arithmetic shows that the edge of our universe is a mere 13,000,000,000 light years from planet Earth. If we had a telescope capable of reaching beyond the 13 billion light years limit, we would see absolutely nothing beyond that point. Any light beyond that point is traveling away from us faster than the speed of light and it would not reach us. However, 1.3 • 1010 light years is quite a respectable distance. Imagine a sphere with 1.3 • 1010 light years as its radius. The formula for the volume of a sphere is Vs=4/3pr3. We want to fill this universe-sphere with protons. Why should we use protons? We use protons because no smaller stable volume exists than that of protons. Electrons are smaller but they lack a stable volume. The volume given to the electron mass seems stable, but electrons have a tendency to become “energy” and reappear again as mass. I define such a behavior as unstable.
The amount of protons that can be swallowed by a sphere of such size is 4.6 • 10124 protons. If you don’t want to accept my numbers by faith, then you could do what I have done: Find the radius of a known atomic nucleus (I used carbon). Count all of its nucleons. Find the mass ratio between protons and neutrons. (Remember that neutrons are a little bigger.) Determine the portion of the nucleus that is occupied by protons. Use this number to get the radius of a single proton. Here are the conclusions: A proton has a volume of 1.7 • 10-40 cubic centimeters, while our universe-sphere has a volume of 7.8 • 1084 cubic centimeters.
A maxitronium is not the number of protons needed to fill the universe — not even close. A maxitronium is the number of different arrangements in which those protons can be combined. The factorial of the total number of protons needed to fill the universe in linear sequences is one maxitronium.
1 maxitronium = 4.6 • 10124! (factorial)
I am not going to solve this factorial number beyond this point. So far, I have defined it for you and I have computed it and calculated it for you. That is as far as I will go. There is probably some naïve recent Ph.D. out there with access to a Pentagon Onyx-19 super computer who, upon reading this article, will want to compute the value of a maxitronium to the last digit. Forget it, kid! You’re going to waste a googol of time and money. No, you could never waste a maxitronium. The temptation for calculating p is a bore because the last digit is unreachable. With the maxitronium, we know that the line has an end. Can you visualize a maxitroniumplex? What about a factorial maxitroniumplex elevated the factorial maxitroniumplex? This sort of reasoning can drive you out of your wits.
If you think that the name maxitronium is as ridiculous as the name googol, keep it to yourself. My second son, Nick, was the person who named the largest number ever. Do ask yourself the following question: “Who is going to use maxitroniums?” Lovers, that’s who. Enamored poets when describing the enormity of their love for YOU!
(b( span="")r /)—The ideas in this article have been copied without permission and without giving me any credit. Please, help me stop plagiarism. Thank you.